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Computational and Applied Mathematics

, Volume 33, Issue 1, pp 193–202 | Cite as

Group analysis of the Novikov equation

  • Yuri Bozhkov
  • Igor Leite FreireEmail author
  • Nail H. Ibragimov
Article

Abstract

We find the Lie point symmetries of the Novikov equation and demonstrate that it is strictly self-adjoint. Using the self-adjointness and the recent technique for constructing conserved vectors associated with symmetries of differential equations, we find the conservation law corresponding to the dilation symmetry and show that other symmetries do not provide nontrivial conservation laws. Then we investigate the invariant solutions.

Keywords

Novikov equation Lie symmetries Strictly self-adjointness  Conservation laws Invariant solutions 

Mathematics Subject Classification (2000)

76M60 58J70 35A30 70G65 

Notes

Acknowledgments

The authors would like to thank FAPESP, São Paulo, Brasil, and BTH, Sweden, for the support giving Nail H. Ibragimov the opportunity to visit IMECC-UNICAMP, where this work was initiated. Yuri Bozhkov would also like to thank FAPESP and CNPq, Brasil, for partial financial support. Igor Leite Freire is thankful to IMECC-UNICAMP for gracious hospitality, UFABC and FAPESP (Grant No. 2011/19089-6) for the financial supports. N. H. Ibragimov’s work is partially supported by the Government of Russian Federation through Resolution No. 220, Agreement No. 11.G34.31.0042.

References

  1. Bluman G (2005) Connections between symmetries and conservation laws. SIGMA 1(paper 011):16Google Scholar
  2. Bluman G, Cheviakov AF, Anco SC (2010) Applications of symmetry methods to partial differential equations. Appl Math Sci 168:398Google Scholar
  3. Bozhkov Y, Silva KAA (2012) Nonlinear self-adjointness of a 2D generalized second order evolution equation. Nonlinear Anal Ser A Theory Appl 75:5069–5078CrossRefzbMATHMathSciNetGoogle Scholar
  4. Camassa R, Holm DD (1993) An integrable shallow water equation with peaked solitons. Phys Rev Lett 71:1661–1664CrossRefzbMATHMathSciNetGoogle Scholar
  5. Degasperis A, Holm DD, Hone ANW (2002) A new integrable equation with peakon solutions. Theor Math Phys 133:1461–1472MathSciNetGoogle Scholar
  6. Freire IL (2012) Conservation laws for self-adjoint first order evolution equation. J Nonlinear Math Phys 18:279–290CrossRefMathSciNetGoogle Scholar
  7. Freire IL (2012) New conservation laws for inviscid Burgers equation. Comput Appl Math 31:559–567CrossRefzbMATHMathSciNetGoogle Scholar
  8. Freire IL (2012) Nonlinear self-adjointness of a generalised fifth-order KdV equation. J Phys A Math Theor 45 (art no 032001)Google Scholar
  9. Freire IL (2013) New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order. Commun Nonlinear Sci Num Simulat 18:493–499CrossRefzbMATHMathSciNetGoogle Scholar
  10. Gandarias ML, Bruzon MS (2012) Some conservation laws for a forced KdV equation. Nonlinear Anal RWA 13:2692–2700CrossRefzbMATHMathSciNetGoogle Scholar
  11. Himonas AA, Holliman C (2012) The Cauchy problem for the Novikov equation. Nonlinearity 25:449–479CrossRefzbMATHMathSciNetGoogle Scholar
  12. Hone ANW, Wang JP (2008) Integrable peakon equations with cubic nonlinearity. J Phys A Math Theor 41(372002):10MathSciNetGoogle Scholar
  13. Hone ANW, Lundmark H, Szmigielski J (2009) Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa–Holm type equation. Dyn Partial Differ Equ 6:253–289CrossRefzbMATHMathSciNetGoogle Scholar
  14. Ibragimov NH (1985) Transformation groups applied to mathematical physics, translated from the Russian mathematics and its applications (Soviet Series), D. Reidel Publishing Co., DordrechtCrossRefGoogle Scholar
  15. Ibragimov NH (2007) A new conservation theorem. J Math Anal Appl 333:311–328CrossRefzbMATHMathSciNetGoogle Scholar
  16. Ibragimov NH, Khamitova RS, Valenti A (2011) Self-adjointness of a generalized Camassa–Holm equation. Appl Math Comp 218:2579–2583CrossRefzbMATHMathSciNetGoogle Scholar
  17. Ibragimov NH (2011) Nonlinear self-adjointness and conservation laws. J Phys A Math Theor 44(432002):8Google Scholar
  18. Ibragimov NH (2011) Nonlinear self-adjointness in constructing conservation laws. Arch ALGA 7/8:1–90 (see also arXiv:1109.1728v1[math-ph], 2011, pp 1–104)Google Scholar
  19. Jiang Z, Ni L (2012) Blow-up phenomenon for the integrable Novikov equation. J Math Anal Appl 385:551–558CrossRefzbMATHMathSciNetGoogle Scholar
  20. Ni L, Zhou Y (2011) Well-posedness and persistence properties for the Novikov equation. J Differ Equ 250:3002–3021CrossRefzbMATHMathSciNetGoogle Scholar
  21. Novikov VS (2009) Generalizations of the Camassa–Holm equation. J Phys A Math Theor 42(342002):14Google Scholar
  22. Olver PJ (1986) Applications of Lie groups to differential equations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  23. Torrisi M, Tracinà R (2013) Quasi self-adjointness of a class of third order nonlinear dispersive equations. Nonlinear Anal RWA 14:1496–1502CrossRefzbMATHGoogle Scholar
  24. Yan W, Li Y, Zhang Y (2012) Global existence and blow-up phenomena for the weakly dissipative Novikov equation. Nonlinear Anal Ser A Theory Appl 75:2464–2473. doi: 10.1016/j.na.2011.10.044 CrossRefzbMATHMathSciNetGoogle Scholar
  25. Wu X, Yin Z (2011) Global weak solutions for the Novikov equation. J Phys A Math Theor 44(055202):17MathSciNetGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013

Authors and Affiliations

  • Yuri Bozhkov
    • 1
  • Igor Leite Freire
    • 2
    Email author
  • Nail H. Ibragimov
    • 3
    • 4
  1. 1.Instituto de Matemática, Estatística e Computação Científica-IMECCUniversidade Estadual de Campinas-UNICAMPCampinasBrasil
  2. 2.Centro de Matemática, Computação e CogniçãoUniversidade Federal do ABC-UFABCSanto AndréBrasil
  3. 3.Laboratory “Group Analysis of Mathematical Models in Natural and Engineering Sciences”Ufa State Aviation Technical UniversityUfaRussia
  4. 4.Department of Mathematics and ScienceBlekinge Institute of TechnologyKarlskronaSweden

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